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The Landauer formula—named after Rolf Landauer, who first suggested its prototype in 1957[1]—is a formula relating the electrical resistance of a quantum conductor to the scattering properties of the conductor.[2] In the simplest case where the system only has two terminals, and the scattering matrix of the conductor does not depend on energy, the formula reads

$$G(\mu )=G_{0}\sum _{n}T_{n}(\mu )\ ,$$

where G is the electrical conductance, $$G_{0}=e^{2}/(\pi \hbar )\approx 7.75\times 10^{{-5}}\Omega ^{{-1}}$$ is the conductance quantum, $$T_{n}$$ are the transmission eigenvalues of the channels, and the sum runs over all transport channels in the conductor. This formula is very simple and physically sensible: The conductance of a nanoscale conductor is given by the sum of all the transmission possibilities that an electron has when propagating with an energy equal to the chemical potential, $$E=\mu$$ .

A generalization of the Landauer formula for multiple probes is the Landauer–Büttiker formula,[3] proposed by Landauer and Markus Büttiker [de]. If probe j has voltage $${\displaystyle V_{j}}$$ (that is, its chemical potential is $${\displaystyle eV_{j}})$$ , and $${\displaystyle T_{i,j}}$$ is the sum of transmission probabilities from probe i to probe j (note that $${\displaystyle T_{i,j}}$$ may or may not equal $${\displaystyle T_{j,i}})$$, the net current leaving probe i i is

$${\displaystyle I_{i}={\frac {e^{2}}{2\pi \hbar }}\sum _{j}\left(T_{j,i}V_{j}-T_{i,j}V_{i}\right)}$$

Ballistic conduction

References

Landauer, R. (1957). "Spatial Variation of Currents and Fields Due to Localized Scatterers in Metallic Conduction". IBM Journal of Research and Development. 1: 223–231. doi:10.1147/rd.13.0223.
Nazarov, Y. V.; Blanter, Ya. M. (2009). Quantum transport: Introduction to Nanoscience. Cambridge University Press. pp. 29–41. ISBN 978-0521832465.
Bestwick, Andrew J. (2015). Quantum Edge Transport in Topological Insulators (Thesis). Stanford University.

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